Download Math 130 Worksheet 2: Linear algebra

Math 130 Worksheet 2: Linear algebra
... but I thought this was a geometry class!
For these exercises, I encourage you to work with a partner. Or two partners!
1. Let E ⊂ F ⊂ K all be fields. (You may assume that they are subfields of C).
Suppose that {a1 , a2 , ...an } is a basis for F as a vector space over E
for K as a vector space over E.
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and {b1 , b2 , ...bm } is a basis
Show that the set of products of pairs {ai bj | 1 ≤ i ≤ n, 1 ≤ j ≤ m} is a basis for K over F .
2. Translate the statement of problem 1 into a statement about degree of field extensions.
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reminder: that means that E is the scalars, and every number x ∈ F can be written as
x = e1 a1 + e2 a2 + ... + en an for some numbers ei ∈ E.